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A computer system has a Weibull failure distribution with acharacteristic life = 97 hours, a shape parameter = 1.2, and alocation parameter equal to 7. When the com...

Question

A computer system has a Weibull failure distribution with acharacteristic life = 97 hours, a shape parameter = 1.2, and alocation parameter equal to 7. When the computer fails it takes anaverage of 7.3 hours to repair.a. Calculate the reliability over a 37-hour period. Show all the work (equations and calculations). (Go to aminimum of 4 decimal places)b. What is the "steady-state" availability of thecomputer system. Show all the work (equations andcalculations). (Go to a minimum of

A computer system has a Weibull failure distribution with a characteristic life = 97 hours, a shape parameter = 1.2, and a location parameter equal to 7. When the computer fails it takes an average of 7.3 hours to repair. a. Calculate the reliability over a 37-hour period. Show all the work (equations and calculations). (Go to a minimum of 4 decimal places) b. What is the "steady-state" availability of the computer system. Show all the work (equations and calculations). (Go to a minimum of 4 decimal places)



Answers

A certain machine has the following hazard function:
$h(t)=\left\{\begin{array}{ll}{.002} & {0<t \leq 200} \\ {.001} & {t>200}\end{array}\right.$
This corresponds to a situation where a device with an exponentially distributed lifetime is
replaced after 200 h of operation by another, better device also having an exponential lifetime
distribution.
(a) Determine and graph the reliability function.
(b) Determine the probability density function of the machine's lifetime
(c) Find the mean time to failure.

Okay, So he told the hours of the wife of the device before it dies, is exponentially distributed to follow some sort of exponential distribution. T is in zero ten. Thirty. They were told that the median number of hours before this divide its dies is five. So that's in for media and five hours. Sorry. Four hours singing this for hours for you. Okay. So again, we want to find this value of a before we can actually do anything with dysfunction. So let's set up our equation for finding the median. T is one half, but here we actually know and is five, four. Yeah, What? I'm thinking fire is four hours. Okay, so we can evaluate this. This is negative. I need to The minus eighty from zero to four is one half too soft for a substance. One minus. Even the marcus forays one has Do you have so hee to the four days to? So a is one forest log, too, Which is about clear point one seven three things. Great. Now we want the probability that the device last longer than five hours. Probably tease greater and equal to five. Well, this is just the inner girl from five to infinity. Ah, here a point one seven thirty three e to the minus your point one seven three three e t. And I'm just going to throw this in my calculator and we get Syria point for two. It's a probability that'LL Last song is by ours.

Yeah. In this exercise, the random variable Y. Is the number of components not staying longer for $1000. So, and it calls for & P. Eco's .8. Using the formula of the phenomenal distribution, we can write the distribution of this random variable, prepared a. We want to find the probability that Because they actually two of the four components last longer than 1000 overs. So this is just a p. two and it echoes point 153, 6 four point B. We want to find out the probability that the sub system operates longer than 1000 Oliver's. So this is just the sum of the probability of p. two, p. 3 and painful. And it echoes point 9 7- eight.

Okay, so we have a random variable. That kid's thie expected life of a computer card Get in by this probability distribution function where tears and ours. So whenever seventy nine t round one half to thirty, two hundred part A, we just want to find the I have a feeling So the most important thing is just to set up the integral. So it's just one half thirty two hundred of tediously rate with t seventy nine hundred t kitty, and what we get is ten eighty. I'm going to risk using a computer out of the system. You can use a graphing calculator whenever you want. Two for being me want Ah, the standard deviation. So first we'll find the parents and the variance is going to be given by this integral. This time we wait with t squared. I mean, I never t minus mu squared When you find this and we just take the square root and then gives us nine fifty two point four five two. The next thing we want to do is find the probability that tea is less than the average minus point one two five times the standard deviation case is just a number. So we'll just set a Senate girls just one half, too. Well, whatever this number is of artist distribution function. And what did we get? We get about zero point five for Teo and then finally your dean, we want Teo find the median. So we find the value in it since that this integral equals one half. And when we saw for M again, she's to computer out of her system. Or you could use a graphing calculator. You get M is equal to six five, six, one over eight, which is eight. Twenty point one two five.

Okay, so we're told that we have and exponentially distributed raining variable, but we're not told about it in this standard way. We're told the media is equal to four. So, uh, if we remember the formula that the median is equal to Ellen of to over Lambda for an exponentially distributed and unbearable, this becomes nice and easy for us to figure out what Lambda is equal to Lambda is equal to Alan off to over four, and then we know that are variable has a pdf of Allen of to over four e to the negative Allen of to next sober for sense of here. We want a product calculate. Probably the component will work without failing for at least five. So we want the 11 minus probability that it fails within five. So that's gonna be one minus and girl from 0 to 5 of our density function. So it's gonna be one minus. Ah, our density function will integrate up to e to the negative Ln of to X over for evaluated from 50 And so, if we plug in our endpoints, we're going to get one minus one minus e to the negative Ellen, too. Times five divided by. So this is kind of a a lot of junk going on. But if we just go ahead and carefully plug it into a calculator, we will get an answer of approximately. Put 4 to 0 for 48 which rounds 2.42 for our final answer.


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